A179412 -id:A179412 - cp's OEIS frontend

A179413 The number of columns with alive cells in the n-th generation of cyclic sequence of patterns given in A179412, played in Conway's Game of Life on the 8x8 toroidal grid.

3, 4, 5, 5, 6, 6, 6, 7, 8, 8, 8, 8, 7, 8, 8, 8, 8, 8, 7, 7, 8, 8, 8, 7, 7, 6, 3, 5, 5, 5, 6, 8, 7, 8, 8, 7, 7, 7, 8, 7, 8, 8, 8, 7, 7, 8, 8, 8, 6, 6, 7, 5, 6, 8, 7, 7, 8, 8, 8, 6, 7, 7, 8, 6, 6, 4, 3, 4, 5, 5, 6, 6, 6, 7, 8, 8, 8, 8, 7, 8, 8, 8, 8, 8, 7, 7, 8, 8, 8, 7, 7, 6, 3, 5, 5, 5, 6, 8, 7

Offset: 0

Antti Karttunen, Jul 27 2010

nonn

Period 66. The sequence begins (from offset 0) with its lexicographically earliest rotation.

The mean value of terms in the whole period of 66 is 6.84848.

			See illustrations in A179412. In the first four patterns there are alive cells in 3, 4, 5, 5 and 5 columns, thus a(0)=3, a(1)=4, and a(2)=a(3)=5.

A. Karttunen, J-program for computing the terms of this sequence

A179414 The number of rows with alive cells in the n-th generation of cyclic sequence of patterns given in A179412, played in Conway's Game of Life on the 8x8 toroidal grid.

Original entry on oeis.org

4, 4, 4, 4, 5, 6, 7, 8, 8, 8, 7, 8, 7, 7, 7, 8, 8, 8, 7, 7, 7, 7, 6, 8, 8, 8, 7, 4, 6, 6, 6, 7, 7, 8, 8, 7, 8, 7, 8, 8, 8, 8, 8, 8, 8, 7, 8, 8, 6, 7, 7, 5, 4, 5, 5, 5, 6, 6, 7, 7, 8, 8, 7, 5, 5, 5, 4, 4, 4, 4, 5, 6, 7, 8, 8, 8, 7, 8, 7, 7, 7, 8, 8, 8, 7, 7, 7, 7, 6, 8, 8, 8, 7, 4, 6, 6, 6, 7, 7

Offset: 0

Antti Karttunen, Jul 27 2010

nonn

Period 66. The sequence begins (from offset 0) with its lexicographically earliest rotation.

The mean value of terms in the whole period of 66 is 6.72727.

			See illustrations in A179412. In all the first four patterns there are alive cells in four rows, thus a(0)=a(1)=a(2)=a(3)=4.

A. Karttunen, J-program for computing the terms of this sequence

A179409 The number of alive cells in Conway's Game of Life on the 8x8 toroidal grid, in a cyclic sequence of 48 patterns containing among other patterns, a "stairstep hexomino" and its mirror image, illustrated below.

Original entry on oeis.org

6, 6, 6, 8, 10, 12, 16, 18, 20, 26, 26, 28, 30, 22, 26, 16, 20, 18, 18, 20, 16, 20, 10, 8, 6, 6, 6, 8, 10, 12, 16, 18, 20, 26, 26, 28, 30, 22, 26, 16, 20, 18, 18, 20, 16, 20, 10, 8, 6, 6, 6, 8, 10, 12, 16, 18, 20, 26, 26, 28, 30, 22, 26, 16, 20, 18, 18, 20, 16, 20, 10, 8

Offset: 0

Antti Karttunen, Jul 27 2010

nonn

Period 24. The sequence begins (from offset 0) with its lexicographically earliest rotation. All terms are even because the initial pattern has an even number of cells and because it has 180-degree rotational symmetry.

The mean value of terms in the whole period of 24 is 16.9167.

			The generations 0-3 of this cycle of patterns look as follows, thus a(0)=a(1)=a(2)=6 and a(3)=8.
. . . . . . . . | . . . . . . . . | . . . . . . . . | . . . . . . . .
. . . . . . . . | . . . . . . . . | . . . . . . . . | . . . . . . . .
. . o . . . . . | . . . . . . . . | . . o . . . . . | . . o o . . . .
. o . . o . . . | . o o o . . . . | . . o o . . . . | . . o . o . . .
. . o . . o . . | . . . o o o . . | . . . o o . . . | . . o . o . . .
. . . . o . . . | . . . . . . . . | . . . . o . . . | . . . o o . . .
. . . . . . . . | . . . . . . . . | . . . . . . . . | . . . . . . . .
. . . . . . . . | . . . . . . . . | . . . . . . . . | . . . . . . . .
(generation 0.) | (generation 1.) | (generation 2.) | (generation 3.)
..................................|stairstep hexomino................
generations 4-7 of this cycle of patterns look as follows, thus a(4)=10, a(5)=12, a(6)=16 and a(7)=18.
. . . . . . . . | . . . . . . . . | . . . . . . . . | . . . . . . . .
. . . . . . . . | . . . . . . . . | . . o . . . . . | . o o . . . . .
. . o o . . . . | . o o o . . . . | . o o o o . . . | . o o . o o . .
. o o . o . . . | . o . . o o . . | o . . . o o . . | o . . . o o . .
. . o . o o . . | . o o . . o . . | . o o . . . o . | . o o . . . o .
. . . o o . . . | . . . o o o . . | . . o o o o . . | . o o . o o . .
. . . . . . . . | . . . . . . . . | . . . . o . . . | . . . . o o . .
. . . . . . . . | . . . . . . . . | . . . . . . . . | . . . . . . . .
In generation 24 we obtain the initial pattern reflected over its central vertical axis, and for the generations 24--47 the patterns repeat the history of the first 24 generations, but reflected over its vertical axis, after which the whole cycle begins from the start again at the generation 48.
. . . . . . . . | . . . . . . . .
. . . . . . . . | . . . . . . . .
. . . . o . . . | . . . . . . . .
. . o . . o . . | . . . o o o . .
. o . . o . . . | . o o o . . . .
. . o . . . . . | . . . . . . . .
. . . . . . . . | . . . . . . . .
. . . . . . . . | . . . . . . . .
(generation 24) | (generation 25)

A. Karttunen, J-program for computing the terms of this sequence
Stephen A. Silver, Life Lexicon, "stairstep hexomino"

Cf. A179410, A179411.
Cf. also A179412 for a longer cyclic sequence of 132 patterns.
A179415 which traces the history of the same initial pattern on infinite square grid, differs from this one for the first time at n=14, where a(14)=26 while A179415(14)=32.

A294241 Longest non-repeating Game of Life on an n X n torus that ends with a fixed pattern.

Original entry on oeis.org

2, 2, 3, 10, 52, 91, 224

Offset: 1

Peter Kagey, Oct 25 2017

We must have a(2n) >= a(n) because one can always place onto a 2n X 2n toroidal board four identical copies of a record-setting pattern for a(n), so that each copy of the pattern "thinks" that it is the sole occupant of an n X n toroidal board and thus acts accordingly. See also comments in A179412 for a related question about the longest repeating pattern on a toroidal board. - Antti Karttunen, Oct 30 2017

			For n = 3 the starting state is:
  +---+---+---+
  | * | * | * |
  +---+---+---+
  |   |   |   |
  +---+---+---+
  |   |   |   |
  +---+---+---+
For n = 4 the starting state is:
  +---+---+---+---+
  | * | * | * |   |
  +---+---+---+---+
  |   |   | * |   |
  +---+---+---+---+
  | * | * |   |   |
  +---+---+---+---+
  |   |   |   |   |
  +---+---+---+---+
For n = 5 the starting state is:
  +---+---+---+---+---+
  | * | * |   | * |   |
  +---+---+---+---+---+
  | * |   |   |   |   |
  +---+---+---+---+---+
  | * | * |   | * | * |
  +---+---+---+---+---+
  | * |   | * |   |   |
  +---+---+---+---+---+
  |   |   |   |   |   |
  +---+---+---+---+---+
From _Bert Dobbelaere_, Jun 20 2024: (Start)
For n = 6 the starting state is:
  +---+---+---+---+---+---+
  |   |   | * |   | * | * |
  +---+---+---+---+---+---+
  | * | * |   |   | * | * |
  +---+---+---+---+---+---+
  |   |   | * |   | * |   |
  +---+---+---+---+---+---+
  | * | * |   |   |   | * |
  +---+---+---+---+---+---+
  |   |   |   |   |   | * |
  +---+---+---+---+---+---+
  |   |   |   | * |   | * |
  +---+---+---+---+---+---+
For n = 7 the starting state is:
  +---+---+---+---+---+---+---+
  | * |   | * | * |   |   | * |
  +---+---+---+---+---+---+---+
  | * |   |   | * |   | * |   |
  +---+---+---+---+---+---+---+
  |   | * |   | * | * |   |   |
  +---+---+---+---+---+---+---+
  |   | * | * |   |   |   | * |
  +---+---+---+---+---+---+---+
  |   | * | * |   | * | * |   |
  +---+---+---+---+---+---+---+
  |   |   |   |   | * |   | * |
  +---+---+---+---+---+---+---+
  |   |   | * |   | * | * |   |
  +---+---+---+---+---+---+---+   (End)

Code Golf Stack Exchange User "Per Alexandersson", Longest non-repeating Game-of-Life sequence

a(7) from Bert Dobbelaere, Jun 20 2024

A370776 The maximum number of alive cells reached in Conway's Game of Life when starting with the first n primes in Ulam's spiral; or -1 if no such maximum exists.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 65, 56, 120, 56, 28, 133, 30, 160, 46, 24, 24, 25, 28, 30, 31, 31, 32, 32, 32, 35, 74, 39, 38, 38, 38, 39, 40, 42, 319, 319, 319, 319, 319, 46, 129, 93, 50, 50, 72, 72, 72, 72, 72, 72, 53, 53, 56, 56, 851, 851, 167, 167, 167, 167, 391

Offset: 1

Thomas Strohmann, Mar 01 2024

nonn

The initial alive cells are at coordinates x=A214664(i), y=A214665(i) for i=1..n.
For the first 7 terms of this sequence we have a(n)=n since those initial configurations do not lead to complex enough patterns that increase the number of alive cells beyond the initial number of alive cells.
The definition includes the possibility that a glider gun (or a similar pattern) is created which will result in an unbounded number of alive cells.

			n=1 to n=4 die out very quickly (within 3 steps). The maximum number of alive cells is simply the number of alive cells in the initial pattern, i.e., n.
n=5 is the first term that leads to somewhat interesting steps in the game of life simulation (although the maximum number of alive cells still does not exceed the initial number 5):
  . . . . . | . . . . . | . . . o . | . . . o . | . . . o . | . . . . .
  o . o . . | . o o o . | . . o . o | . . o . o | . . . o . | . . . . .
  . . o o . | . . o o . | . . o . o | . . . . . | . . . . . | . . . . .
  o . . . . | . . . . . | . . . . . | . . . . . | . . . . . | . . . . .
n=8 leads to a maximum number of 65 alive cells and stabilizes after 107 steps. Initial pattern:
  o . . . o |
  . o . o . |
  o . . o o |
  . o . . . |
n=15 reaches a maximum of 160 alive cells and is the first pattern that leads to having a glider (escaping in the northeast direction). Besides the glider, the stabilized pattern contains 4 blinkers, 3 blocks, 2 beehives and 1 ship.

Wikipedia, Ulam spiral.
Wikipedia, Conway's Game of Life.

Cf. A214664, A214665.

Cf. A152389, A126803, A099733, A179412.

cp's OEIS Frontend

A179413 The number of columns with alive cells in the n-th generation of cyclic sequence of patterns given in A179412, played in Conway's Game of Life on the 8x8 toroidal grid.

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A179414 The number of rows with alive cells in the n-th generation of cyclic sequence of patterns given in A179412, played in Conway's Game of Life on the 8x8 toroidal grid.

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A179409 The number of alive cells in Conway's Game of Life on the 8x8 toroidal grid, in a cyclic sequence of 48 patterns containing among other patterns, a "stairstep hexomino" and its mirror image, illustrated below.

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A294241 Longest non-repeating Game of Life on an n X n torus that ends with a fixed pattern.

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A370776 The maximum number of alive cells reached in Conway's Game of Life when starting with the first n primes in Ulam's spiral; or -1 if no such maximum exists.

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